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I am playing with Sequent Calculus(Just started studying it) and I want to know how to express the Syllogism:

Socrates is a Man.
All Men are Mortal.
Therefore Socrates is a Mortal.

In terms of Sequent Calculus.

My attempt so far is as follows:

1.

There is a set Mortals and Men, so the following statement should be axiomatically true. $$ \overline{\Gamma, Mortals, Men \vdash Mortals, Men} $$

Which is the same as saying

$$ \overline{\Gamma, Mortals, Men} $$

2.

"All Men are Mortals" is a part of my assumptions, so the following statement seems to express it but I am having doubts.

$$ \overline{\Gamma, Mortals \rightarrow Men} $$

3.

"Socrates is a Mortal" is another assumption that must be a part of the environment.

$$ \overline{\Gamma, Mortals \rightarrow Socrates} $$

This looks weird, since it looks like I'm saying that all mortals are socrates, but I am trying to say that Socrates is in Mortals(by turning the turnstile into implication), but I am not sure whether $\vdash$ is the same as $\rightarrow$ except can be nested inside a sequent, or if there's another way to go around expressing this.

4. Finally, I want to say that if Socrates is in my environment, then Socrates is a Mortal,

$$ \overline{\Gamma, Socrates} $$

5.

but at this point I have no idea what's going on, but Ill try to put it together anyway:

$$ \overline{\Gamma, Socrates \vdash Mortal}\\ \overline{\Gamma, Men \rightarrow Socrates, Mortal \rightarrow Men} $$

Can somebody look over what I did and help me gain some confidence in how to express it correctly(If it is possible to express it in Sequent Calculus)?

1 Answers 1

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It seems to me that we need this:

1) $Man(S) \to Man(S)$ --- top left

2) $Mortal(S) \to Mortal(S)$ --- top right

3) $Man(S) \supset Mortal(S), Man(S) \to Mortal(S)$ --- from 1) and 2) by $\supset$-left

4) $\forall x(Man(x) \supset Mortal(x)), Man(S) \to Mortal(S)$ --- from 3) by $\forall$-left

5) $\to \forall x(Man(x) \supset Mortal(x))$ --- "all men are mortals"

6) $Man(S) \to Mortal(S)$ --- from 4) and 5) by Cut

7) $\to Man(S)$ --- "Socrates is a man"

8) $\to Mortal(S)$ --- from 6) and 7) by Cut : "Socrates is mortal".

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    is the function/lambda notation necessary? or is there an introduction rule which lets us expand it? and what's the difference between the arrow and the flipped c. – 2017-02-21
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    @Dmitry - the syntax of seq calculus is : $\Gamma \to \Delta$. This needs $\supset$ for the conditional. With your notation : $\Gamma \vdash \Delta$ we have freed $\to$ to be used for the conditional. – 2017-02-21
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    What is "function/lambda notation" ? We are working with predicate logic : $\forall x Man(x)$ and $Man(Socrates)$. – 2017-02-21
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    Can this predicate notation be reduced to an equivalent statement in sequent calculus? Or is it a necessary primitive in sequent calculus for expressing labelling? – 2017-02-21
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    @Dmitry - sorry... I do not understand. But if you are meaning that you are trying to prove it with propositional calculus, then you cannot prove it that way. – 2017-02-21
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    I am wondering if the notation Man(x) can be reduced to a more basic sequent calculus expressions I am familiar with, or whether I have to accept it as a primitive without an introduction. – 2017-02-21